Linear Operators: Spectral theory |
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Page 898
If we put E ( d ) = 0 when d n o ( T ) is void , then Corollary 4 follows immediately from Theorem 1 and Corollary IX.3.15 . Q.E.D. 5 DEFINITION . The uniquely defined spectral measure associat5 ed , in Corollary 4 , with the normal ...
If we put E ( d ) = 0 when d n o ( T ) is void , then Corollary 4 follows immediately from Theorem 1 and Corollary IX.3.15 . Q.E.D. 5 DEFINITION . The uniquely defined spectral measure associat5 ed , in Corollary 4 , with the normal ...
Page 1301
However , as vo v6 6 satisfies an equation of order 2n , v , must be identically zero . This contradiction completes the proof . Q.E.D. 23 COROLLARY . Lett be a formal differential operator of order n on an interval I with end points a ...
However , as vo v6 6 satisfies an equation of order 2n , v , must be identically zero . This contradiction completes the proof . Q.E.D. 23 COROLLARY . Lett be a formal differential operator of order n on an interval I with end points a ...
Page 1459
Q.E.D. urma 30 COROLLARY . A formally positive formally symmetric formal differential operator r is finite below zero . PROOF . It is obvious from Definition 20 that t is bounded below . τ Thus the present corollary follows from ...
Q.E.D. urma 30 COROLLARY . A formally positive formally symmetric formal differential operator r is finite below zero . PROOF . It is obvious from Definition 20 that t is bounded below . τ Thus the present corollary follows from ...
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extensive presentation of applications of the spectral theorem | 911 |
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additive adjoint adjoint operator algebra analytic assume basis belongs Borel set boundary conditions boundary values bounded called clear closed closure commutative compact complex Consequently consider constant contains converges Corollary corresponding defined Definition denote dense determined domain eigenvalues elements equal equation equivalent Exercise exists extension fact finite dimensional follows follows from Lemma formal differential operator formula function given Hence Hilbert space Hilbert-Schmidt ideal identity immediately implies independent inequality integral interval invariant isometric isomorphism Lemma limit linear Ly(R mapping matrix measure multiplicity neighborhood norm obtained orthonormal positive preceding present projection proof properties prove range regular representation respectively restriction result satisfies seen sequence shown shows solutions spectral spectrum square-integrable statement subset subspace sufficient Suppose symmetric Theorem theory topology transform uniformly unique unit unitary vanishes vector zero